System and method for adaptive reduced-rank parameter estimation using an adaptive decimation and interpolation scheme

ABSTRACT

The present invention describes a system and method for general parameter estimation using adaptive processing that provides a performance that significantly exceeds existing reduced-rank schemes using reduced computational resources with greater flexibility. The adaptive processing is accomplished by calculating a reduced-rank approximation of an observation data vector using an adaptive decimation and interpolation scheme. The new scheme employs a time-varying interpolator finite impulse response (FIR) filter at the front-end followed by a decimation structure that processes the data according to the decimation pattern that minimizes the squared norm of the error signal and by a reduced-rank FIR filter. According to the present invention, the number of elements for estimation is substantially reduced, resulting in considerable computational savings and very fast convergence performance for tracking dynamic signals. The current invention is aimed at communications and signal processing applications such as equalization, interference suppression of CDMA systems, echo cancellation and beamforming with antenna arrays. Amongst other promising areas for the deployment of the present technique, we also envisage biomedical engineering, control systems, radar and sonar, seismology, remote sensing and instrumentation.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a system and method for improving theperformance of estimation techniques and for reducing the number ofelements required for this estimation task when dealing with generalpurpose signals that are embedded in interference.

2. Discussion of the Related Art

Several signals processing techniques have been developed to processsignals and estimate parameters of interest based on the knowledge of areference signal. Efforts are generally made to reduce or suppress theinterference received with the signal and this interference can be ofvarious types such as noise, jamming and other users. When these signalspossess a large bandwidth and require a large number of elements forestimation, the signal processing task becomes rather challenging.

Adaptive signal processing is particular technique designed to model,extract and track signals by weighting a set of discrete-time signalsfrom a source, which can be an antenna or a general electronicequipment, in order to perform a desired task.

To compute the adaptive weights, these techniques typically combineseveral samples over a period of time. Generally, adaptive weights arecalculated through the relationship Rw=p, where p is the steering vectorwith M coefficients, R is M×M the covariance matrix, and w is the weightvector with M elements. In order to identify the adaptive weights thisrelationship is simply manipulated to the following: w=R⁻¹p. Thisequation requires a number of arithmetic operations that is proportionalto M³, which is too complex for practical use.

In this context, existing adaptive signal processing techniques such astransversal linear filters with the least-mean square (LMS) algorithmare simple, have low complexity but usually have poor convergenceperformance. In contrast, adaptive filters with recursive least-squares(RLS) algorithms have fast convergence but require a significantlyhigher complexity than LMS recursions. Several attempts to providecost-effective parameter estimators with fast convergence performancehave been made in the last few decades through variable step sizealgorithms, data-reusing, averaging methods, sub-band andfrequency-domain adaptive filters and RLS types algorithms with linearcomplexity such as lattice-based implementations, fast RLS algorithms,QR-decomposition-based RLS techniques and the more recent and promisingreduced-rank adaptive filters.

The advantages of reduced-rank adaptive filters are their fasterconvergence speed and better tracking performance over existingtechniques when dealing with large number of weights. Variousreduced-rank methods and systems were based on principal componentsanalysis, in which a computationally expensive singular valuedecomposition (SVD) to extract the signal subspace is required. Otherrecent techniques such as the multistage Wiener filter (MWF) ofGoldstein et al. in “A multistage representation of the Wiener filterbased on orthogonal projections”, IEEE Transactions of InformationTheory, vol. 44, November, 1998—perform orthogonal decompositions inorder to compute its parameters, leading to very good performance and acomplexity inferior to those systems that require poor performance insystems with moderate to heavy loads.

In most applications, the process of calculating and altering theweights must be done in real-time. Because modern applications involve alarge number of adaptive parameters and operate in non-stationaryenvironments, the system requires a large amount of data to compute theestimates. However, in most practical situations the amount of dataavailable is simply insufficient to provide accurate estimates. Inaddition, when a parameter estimator with a large number of weights isrequired to track a dynamic signal embedded in interference, itencounters difficulties in following the signal of interest and may fallor show unsatisfactory performance.

These and other deficiencies exist in current adaptive processingsystems in the open literature and amongst the patented techniques sofar. Therefore, a solution to these problems is needed providing areduced rank adaptive processing system and method specifically designedto more accurately estimate and track signals that involve a largenumber of processing elements with low complexity and great flexibility.

BRIEF DESCRIPTION OF THE FIGURES

For a better understanding of the present invention, reference may bemade to the accompanying figures.

FIG. 1 depicts the proposed adaptive reduced-rank filter structure withadaptive decimation and interpolation.

FIG. 2 illustrates the process of sample selection of the three proposedadaptive decimation schemes for a system with M=8 samples, anddecimation factor L=4 with B=4 branches where (a) is the optimaldecimation procedure; (b) is the random decimation; and (c) is thepre-stored uniform decimation patterns.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

The present invention relates to a reduced-rank parameter estimatorsystem and method for processing discretely represented data based on anovel adaptive interpolation and decimation scheme that is simple,flexible, and provides a remarkable performance advantage over existingtechniques. The new scheme employs a time-varying interpolator FIRfilter at the front-end followed by a decimation structure thatprocesses the data according to the decimation pattern that minimizesthe squared norm of the error signal between the desired estimate andthe estimate provided by a reduced-rank filter. In the new scheme, thenumber of elements for estimation is substantially reduced, resulting inconsiderable computational savings and very fast convergence performancefor tracking dynamic signals. Minimum mean squared error (MMSE) designfilters for both interpolator and reduced-rank estimators is describedand it is proposed alternative decimation structures for the proposedscheme. With respect to the decimation structures the invention definesthe optimal decimation scheme that represents a combinatorial problemwith all possible decimation patterns for a chosen decimation factor Land two sub-optimal decimation structures that are based on random andpre-stored patterns, respectively. In order to further reduce thecomputational burden for parameter estimation, it is illustrated howstandard low complexity LMS and RLS algorithms can be used inconjunction with the presented reduced-rank structure. The presentinvention is med at communications and signal processing applicationssuch as equalization, interference suppression for CDMA systems, echocancellation and beamforming with antenna arrays. Amongst otherpromising areas for the deployment of the present technique, it is alsoenvisaged biomedical engineering, control systems, radar and sonar,seismology, remote sensing and instrumentation.

1. Reduced-Rank MMSE Filtering with Adaptive Interpolation andDecimation

The framework of the proposed adaptive reduced-rank MMSE filteringscheme and method is detailed in this section. FIG. 1 shows thestructure of the adaptive processor, where an interpolator, a decimatorunit and a reduced-rank receiver that are time-varying are employed. TheM×1 received vector r(i)=[r_(O) ^((i)) . . . r_(M−1) ^((i))]^(T) isfiltered by the interpolator filter v(i)=[v₀ ^((i)) . . . v_(NI−1)^((i))]^(T), yielding the interpolated received vector r_(I)(i), whichis then decimated by several decimation patterns in parallel, leading toB different M/L×1-dimensional vectors r_(b)(i). The novel decimationprocedure corresponds to discard M−M/L samples of r_(I)(i) of each setof M received samples with different discarding patterns, resulting in Bdifferent decimated vectors r_(b)(i) with reduced dimension M/L and thencomputing the inner product of r_(b)(i) with the M/L-dimensional vectorof the reduced-rank filter coefficients w(i)=[w_(O) ^((i)) . . . w_(M/L)^((i)]) ^(T) that minimizes the squared norm of the error signal.

2. Adaptive Interpolation and Decimation Structure

The front-end adaptive filtering operation is carried out by theinterpolator filter v(i) on the received vector r(i) and yields theinterpolated received vector r_(I)(i)=V^(H)(i)r(i), where the M×Mconvolution matrix V(i) with the coefficients of the interpolator isgiven by

$\begin{matrix}{{V(i)} = \begin{bmatrix}v_{0}^{(i)} & \ldots & v_{N_{I} - 1}^{(i)} & 0 & \ldots & 0 \\\vdots & \vdots & \vdots & ⋰ & ⋰ & \vdots \\0 & \ldots & 0 & v_{0}^{(i)} & \ldots & v_{N_{I} - 1}^{(i)}\end{bmatrix}} & (1)\end{matrix}$

An alternative way of expressing the interpolated received vectorr_(I)(i) is now introduced that will be useful when dealing with thedifferent decimation patterns, through the following equivalence:r _(I)(i)=V ^(H)(i)r(i)=

(i)v*(i)  (2)where the M×N_(I) matrix with the received samples of r(i) and thatimplements convolution is described by

$\begin{matrix}{{{\underset{\_}{\Re}}_{o}(i)} = \begin{bmatrix}r_{0}^{(i)} & r_{1}^{(i)} & \ldots & r_{N_{I} - 1}^{(i)} \\r_{1}^{(i)} & r_{2}^{(i)} & \ldots & r_{N_{I}}^{(i)} \\\vdots & \vdots & ⋰ & \vdots \\r_{M - 1}^{(i)} & r_{M}^{(i)} & \ldots & r_{M + N_{I}}^{(i)}\end{bmatrix}} & (3)\end{matrix}$

The decimated interpolated observation vector r _(b)(i)=D_(b)r_(I)(i)for branch b is obtained with the aid of the M/L×M decimation matrixD_(b) that adaptively minimizes the squared norm of the error at timeinstant i. The matrix D_(b) is mathematically equivalent to signaldecimation with a chosen pattern on the M×1 vector r_(I)(i), whichcorresponds to the removal of M−M/L samples of r_(I)(i) of each set of Mobserved samples. An interpolated and decimation scheme with uniformdecimation pattern D_(b) ^(U) and with decimation factor L can bedesigned by choosing the number of branches B=1 and the structure:

$\begin{matrix}{D_{b}^{U} = \begin{bmatrix}\begin{matrix}\begin{matrix}1 & {\; 0} & {\; 0} & {\; 0} & {\; 0} & \ldots & 0 & 0 & 0 & 0 & 0 \\\vdots & {\;\vdots} & {\;\vdots} & {\;\vdots} & {\;\vdots} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\end{matrix} \\\begin{matrix}\underset{\underset{{({m - 1})}L\mspace{11mu}{zeros}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}} & 1 & 0 & \ldots & 0 & 0 & 0 & 0 & 0\end{matrix}\end{matrix} \\\begin{matrix}\underset{\underset{{({{M/L} - 1})}L\mspace{11mu}{zeros}}{︸}}{\begin{matrix}\begin{matrix}0 & 0 & 0\end{matrix} & 0 & 0 & \ldots & 0\end{matrix}} & 1 & \underset{\underset{L - {1\mspace{11mu}{zeros}}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}}\end{matrix}\end{bmatrix}} & (4)\end{matrix}$where m (m=1, 2, . . . , M/L) denotes the m-th row. The uniformdecimation pattern corresponds to the use of a single branch on thedecimation unit. However, it is possible to exploit the processedsamples through a more elegant and effective way with the deployment ofseveral branches in parallel. Specifically, the next subsection willpresent alternatives for designing the decimation unit that can yield aremarkable performance and allow the deployment of large decimationfactors L.3. Adaptive Decimation Schemes

In this subsection, it is proposed three alternatives for designing thedecimation unit of the novel reduced-rank scheme, where the commonframework is the use of parallel branches with decimation patterns thatyield B decimated vectors r _(b)(i) as candidates. The first structurecorresponds to the optimal decimation procedure for a given decimationfactor L that seeks the solution to the decimation problem thatadaptively minimizes the square norm of the error. The second approachis a suboptimal one that utilizes random decimation patterns, whereasthe third scheme employs pre-stored patterns in lieu of the random ones.A graphical illustration of the three proposed decimation patterns isshown in FIG. 2, considering small M=8, L=4 and B=4 for simplicity andwith the aim of clearly explaining the adaptive decimation process.

Mathematically, the signal selection scheme chooses the decimationpattern D_(b) and consequently the decimated interpolated observationvector r _(b)(i) that minimize |e_(b)(i)|², where e_(b)(i)=d(i)−w^(H)(i)r _(b)(i)=w^(H)(i)D_(b)r_(I)(i) is the error signal at branch b. Oncethe decimation pattern is selected for the time instant i, the decimatedinterpolated vector is computed as follows r(i)=D(i)r_(I)(i). Thedecimation pattern D(i) is selected on the basis of the followingcriterion:

$\begin{matrix}{{D(i)} = {\arg\mspace{11mu}{\min\limits_{1 \leq b \leq B}{{e_{b}(i)}}^{2}}}} & (5)\end{matrix}$where the optimal decimation pattern D_(opt) for the proposed schemewith decimation factor L is derived through the counting principle,where it is considered a procedure that has M samples as possiblecandidates for the first row of D_(opt) and M-m samples as candidatesfor the following M/L−1 rows of D_(opt), where m denotes the m-th row ofthe matrix D_(opt), resulting in a number of candidates equal to

$\begin{matrix}{B = {\underset{{M/L}\mspace{11mu}{terms}}{\underset{︸}{{M( {M - 1} )}( {M - 2} )\mspace{11mu}\ldots\mspace{11mu}( {M - {M/L} - 1} )}} = \frac{M!}{( {M - {M/L}} )!}}} & (6)\end{matrix}$

The optimal decimation scheme described in (5)-(6) is, however, verycomplex for practical use because it requires the M/L permutation of Msamples for each symbol interval and carries out an extensive searchover all possible patterns. Therefore, a decimation scheme that rendersitself to practical and low complexity implementations is of greatsignificance and interest in this context.

The second decimation scheme that it is presented in this subsection isa suboptimal approach that is based on a finite number of parallelbranches with random decimation patterns D_(b) ^(R), where 1≦b≦B andwhose structure is described by

$\begin{matrix}{D_{b}^{R} = \begin{bmatrix}\begin{matrix}\underset{\underset{r_{1}\;{zeros}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}} & 1 & 0 & \ldots & 0 & 0 & 0 & 0 & 0\end{matrix} \\\begin{matrix}{\vdots\mspace{14mu}} & \vdots & {\mspace{11mu}\vdots\mspace{11mu}} & {\vdots\mspace{11mu}} & \vdots & {\mspace{11mu}\vdots\mspace{11mu}} & {\vdots\mspace{11mu}} & {\vdots\mspace{11mu}} & {\vdots\;} & {\;\vdots\;} & \vdots\end{matrix} \\\begin{matrix}\underset{\underset{r_{m}\;{zeros}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}} & 1 & 0 & \ldots & 0 & 0 & 0 & 0 & 0\end{matrix} \\\begin{matrix}\underset{\underset{\frac{r_{M}}{L}\;{zeros}}{︸}}{\begin{matrix}0 & 0 & 0 & 0 & 0 & \ldots & 0\end{matrix}} & 1 & \underset{\underset{{({M - \frac{r_{M}}{L} - 1})}\;{zeros}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}}\end{matrix}\end{bmatrix}} & (7)\end{matrix}$where m(m=1, 2, . . . , M/L) denotes the m-th row and r_(m) is adiscrete uniform random variable, which is independent for each row mand whose values range between 0 and M−1. The disadvantage of the abovedecimation pattern is that it requires the use of a discrete uniformrandom generator for producing the B decimation patterns which areemployed in parallel. In this regard, the r_(m) does not have to benecessarily changed for each time instant i, but it can be used for thewhole set of data.

The third decimation scheme that it is introduced in this invention is asuboptimal structure that employs B pre-stored decimation patterns whosestructure is given by

$\begin{matrix}{D_{b}^{U} = \begin{bmatrix}\begin{matrix}\underset{\underset{{({b - 1})}\mspace{11mu}{zeros}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}} & 1 & 0 & \ldots & 0 & 0 & 0 & 0 & 0\end{matrix} \\\begin{matrix}{\vdots\mspace{14mu}} & \vdots & {\mspace{11mu}\vdots\mspace{11mu}} & {\vdots\mspace{11mu}} & \vdots & {\mspace{11mu}\vdots\mspace{11mu}} & {\vdots\mspace{11mu}} & {\vdots\mspace{11mu}} & {\vdots\;} & {\;\vdots\;} & \vdots\end{matrix} \\\begin{matrix}\underset{\underset{{{({m - 1})}L} + {{({b - 1})}\;{zeros}}}{︸}}{\begin{matrix}\begin{matrix}{0\;} & {0\;} & {0\;}\end{matrix} & {\ldots\;} & {0\;}\end{matrix}} & {0\;} & 1 & 0 & 0 & 0 & 0\end{matrix} \\\begin{matrix}\underset{\underset{{{({{M/L} - 1})}L} + {{({b - 1})}\mspace{11mu}{zeros}}}{︸}}{\begin{matrix}0 & 0 & 0 & 0 & 0 & \ldots & 0\end{matrix}} & 1 & \underset{\underset{{{({L - 1})} - {({b - 1})}}\;{zeros}}{︸}}{\begin{matrix}0 & \ldots & 0\end{matrix}}\end{matrix}\end{bmatrix}} & (8)\end{matrix}$where in the above structure the designer utilizes uniform decimationfor each branch b and the different patterns are obtained by picking upadjacent samples with respect to the previous and succeeding decimationpatterns. In fact, the proposed decimation structure corresponds to auniform scheme whose index of samples chosen at each row m is shifted byone position for each branch b. The great advantage of the abovedecimation pattern is that it is very simple and can be easilyimplemented by digital signal processors since it is constituted bysliding patterns in parallel. It should be remarked that othersuboptimal decimation structures have been tested even though theschemes described here, namely, the random structure D_(b) ^(R) and thepre-stored one D_(b) ^(S) have shown the best results and are able toapproach the performance of the optimal scheme D_(opt).4. MMSE Reduced-Rank Scheme Filter Design

In this subsection, it is defined the MMSE filter design of the proposedreduced-rank structure. The strategy, that allows solutions devise forboth interpolator and receiver, is to express the estimated symbolx(i)=w^(H)(i) r(i) as a function of w(i) and v(i):

$\quad\begin{matrix}\begin{matrix}{{x(i)} = {{{w^{H}(i)}{\overset{\_}{r}(i)}} = {{{w^{H}(i)}( {{D(i)}{r_{I}(i)}} )} = {{w^{H}(i)}( {{D(i)}{V(i)}{r(i)}} )}}}} \\{= {{{w^{H}(i)}( {{D(i)}{\Re_{o}(i)}{v^{*}(i)}} )} = {{w^{H}(i)}( {{D(i)}{\Re_{o}(i)}} ){v^{*}(i)}}}} \\{= {{w^{H}(i)}{\Re(i)}{v^{*}(i)}}} \\{= {{{v^{H}(i)}( {{\Re^{T}(i)}{w^{*}(i)}} )} = {{v^{H}(i)}{u(i)}}}}\end{matrix} & (9)\end{matrix}$where u(i)=

^(T)(i)w*(i) is an N_(I)×1 vector, the M/L coefficients of w(i) and theN_(I) elements of v(i) are assumed to be complex and the M/L×N_(I)matrix

(i) is given by

(i)=D(i)

_(O)(i).

The MMSE solution for w(i) and v(i) can be computed if it is consideredthe optimization problem whose cost function is expressed byJ _(MSE)(w(i),v(i))=E[|d(i)−v ^(H)(i)

^(T)(i)w*(i)|²]  (10)where d(i) is the desired symbol at time index (i). By fixing theinterpolator v(i) and minimizing (10) with respect to w(i) theinterpolated Wiener filter weight vector isw(i)=α(v)= R ⁻¹(i) p (i)  (11)where R(i)=E[ r(i) r ^(H)(i)], p(i)=E[d*(i) r(i)], r(i)=

^(T)(i)v*(i) and by fixing w(i) and minimizing (10) with respect to v(i)the interpolator weight vector isv(i)=β(w)= R _(u) ⁻¹(i) p _(u)(i)  (12)where R _(u)(i)=E[u(i)u^(H)(i)], p _(u)(i)=E[d*(i)u(i)] and u(i)=

^(T)(i)w*(i). The associated MSE expression areJ(v)=J _(MSE)(v,α(v))=σ_(d) ² − p ^(H)(i) R ⁻¹(i) p (i)  (13)J _(MSE)(w,β(w))=σ_(d) ² − p _(u) ^(H)(i) R _(u) ⁻¹(i) p _(u)(i)  (14)where σ_(d) ²=E[|d(i)|²]. Note that points of global minimum of (10) canbe obtained by

$v_{opt} = {\arg\mspace{11mu}{\min\limits_{v}{{J(v)}\mspace{14mu}{and}}}}$w_(opt) = α(v_(opt))  or$w_{opt} = {\arg\mspace{11mu}{\min\limits_{w}{{J_{MSE}( {w,{\beta(w)}} )}\mspace{14mu}{and}}}}$v_(opt) = β(w_(opt)).

At the minimum point (13) equals (14) and the MMSE for the proposedstructure is achieved. It is important to remark that (11) and (12) arenot closed-form solutions for w(i) and v(i) since (11) is a function ofv(i) and (12) depends on w(i) and thus it is necessary to iterate (11)and (12) with an initial guess to obtain a solution. A pseudo-code forthe batch iterative algorithm is shown in Table I. An alternativeiterative MMSE solution can be seek via adaptive algorithms and isdiscussed in the next section.

TABLE I Batch iterative algorithm for the design of proposedreduced-rank filtering scheme. Algorithm 1: Initialize w(0) = [0 . . .0]^(T) end v(0) = [1 0 . . . 0]^(T) Choose parameters B and L for eachtime instant (i) do Compute M × 1 vector r_(I)(i) = V^(H) (i)r(i) =

_(o) (i)v*(i) Obtain M/L × 1 vectors r _(b)(i) = D_(b)r_(I)(i) for the Bbranches Select D_(b) that minimizes |e_(b)(i)|² (D_(b) becomes D(i))Obtain r(i) and compute estimate of R(i) and p(i)$( {{{Ex}\text{:}\mspace{11mu}{\hat{\overset{\_}{R}}(i)}} = {{\sum\limits_{j = 1}^{i}{\lambda^{i - j}{r(j)}{r^{H}(j)}\mspace{20mu}{and}\mspace{14mu}{\hat{\overset{\_}{p}}(i)}}} = {\sum\limits_{j = 1}^{i}{\lambda^{i - j}{d^{*}(i)}{\overset{\_}{r}(j)}}}}} )$Obtain u(i) =

^(T) (i)w*(i) and compute estimates of R _(u)(i) and p _(u)(i)${{Compute}\mspace{20mu}{v(i)}} = {{{{\hat{\overset{\_}{R}}}_{u}^{- 1}(i)}{{\hat{\overset{\_}{p}}}_{u}(i)}\mspace{20mu}{and}\mspace{20mu}{w(i)}} = {{{\hat{\overset{\_}{R}}}^{- 1}(i)}{\hat{\overset{\_}{p}}(i)}}}$Reduced-rank sample estimate: {circumflex over (d)}(i) = w^(H) (i)r(i)

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the spirit or scope of the invention. Thus, it isintended that the present invention cover the modifications andvariations of this invention provided that they come within the scope ofany claims and their equivalents.

5. Adaptive Algorithms

Here it is described alternative forms of computing the parameters ofthe invented system and method. This estimation procedure isaccomplished by standard LMS and RLS algorithms commonly found in theliterature that are slightly modified in order to account for the jointoptimization of both interpolator and reduced-rank filters. Thesealgorithms are used to adjust the parameters of the reduced-rank and theinterpolator filters based on the minimum squared error (MSE) criterionand select the decimation pattern that minimizes the squared norm of theerror signal and require fewer computational resources than the batchapproach described in Table I. The novel system and method, shown inFIG. 1, gathers fast convergence, low complexity and additionalflexibility since the designer can adjust the decimation factor L andthe length of the interpolator N_(I) depending on the needs of theapplication and the hostility of the environment.

5. 1. Least Means Squares (LMS) Algorithm for the Proposed Scheme

Let's consider the observation vector r(i) and the adaptive processingcarried out by the proposed scheme, as in FIG. 1. With the aid of theconvolution matrix in (1), it is computed the M×1 interpolatedobservation vector r_(I)(i) and then computed the decimated interpolatedobservation vectors r _(b)(i) for the B branches with the aid of thedecimation patterns D_(b), where 1≦b≦B. Once the B candidate vectors r_(b)(i) are computed, it is selected the vector r _(b)(i) that minimizesthe squared norm ofe _(b)(i)=d(i)−w ^(H)(i) r _(b)(i)  (15)

Based on the signal selection that minimizes |e_(b)(i)|², thecorresponding reduced-rank observation vector r(i) is chosen and theerror of the proposed LMS algorithm e(i) is selected as the errore_(b)(i) with smallest squared magnitude of the B branches

$\begin{matrix}{{e(i)} = {\arg\mspace{11mu}{\min\limits_{1 \leq b \leq B}{{e_{b}(i)}}^{2}}}} & (16)\end{matrix}$

Given the reduced-rank observation vector r(i) and the desired signald(i), the following cost function is done as:J _(MSE)(w(i),v(i)=|d(i)−v ^(H)(i)

^(T)(i)w*(i)|²  (17)

Taking the gradient terms of (17) with respect to v(i), w(i) and usingthe gradient descent

$\begin{matrix}{{{w( {i + 1} )} = {{w(i)} - {\mu\frac{\partial{J_{MSE}( {{w(i)},{v(i)}} )}}{\partial w^{*}}\mspace{14mu}{and}}}}\text{}{{v( {i + 1} )} = {{v(i)} - {\eta\frac{\partial{J_{MSE}( {{w(i)},{v(i)}} )}}{\partial v^{*}}\mspace{20mu}{yields}\text{:}}}}} & \; \\{{v( {i + 1} )} = {{v(i)} + {\eta\;{e^{*}(i)}{u(i)}}}} & (18) \\{{w( {i + 1} )} = {{w(i)} + {\mu\;{e^{*}(i)}{r(i)}}}} & (19)\end{matrix}$where e(i)=d(i)−w^(H)(i) r(i), u(i)=

^(T)(i)w*(i), μ and η are the step sizes of the algorithm for w(i) andv(i). The LMS algorithm for the proposed structure described in thissection has a computational complexity O(M/L+N_(I)). In fact, theproposed structure trades off one LMS algorithm with complexity O(M)against two LMS algorithms with complexity O(M/L) and O(N_(I)),operating in parallel. It is worth noting that, for stability and tofacilitate tuning of parameters, it is useful to employ normalized stepsizes and consequently NLMS type recursions when operating in a changingenvironment and thus have μ(i)=μ_(O)/(r^(H)(i)r(i)) andη(i)=η_(O)/(u^(H)(i)u(i) as the step sizes of the algorithm for w(i) andv(i), where μ_(O) and η_(O) are their respective convergence factors.

TABLE 2 LMS iterative algorithm for the design of proposed reduced-rankfiltering scheme. Algorithm II: Initialize w(0) = [0 . . . 0]^(T) andv(0) = [1 0 . . . 0]^(T) Choose parameters B, L and step sizes μ and ηfor each time instant (i) do Compute M × 1 vector r_(I) (i) = V^(H)(i)r(i) =

_(o) (i)v* (i) Obtain M/L × 1 vectors r _(b) (i) = D_(b)r_(I) (i) forthe B branches Select D_(b) that minimizes |e_(b)(i)|² (D_(b) becomesD(i)) Obtain   r(i),  u(i)  and  compute  v(i + 1) = v(i) + ηe* (i)u(i)and w(i + 1) = w(i) + μe* (i)r(i) Reduced-rank sample estimate:{circumflex over (d)}(i) = w^(H) (i)r(i)5. 2. Recursive Least Squares (RLS) Algorithm for the Proposed Scheme

Now let's consider again the observation vector r(i) and the adaptiveprocessing carried out by the proposed scheme, as depicted in FIG. 1. Itis computed the M×1 interpolated observation vector r_(I)(i) with theaid of V(i) and then the decimated interpolated observation vectorsr_(b)(i) is computed, for the B branches with the decimation patternsD_(b), where 1≦b≦B. Unlike the LMS algorithms presented in the precedingsubsection, the vector r _(b)(i) that minimizes the squared norm ischosen of the posteriori errorζ_(b)(i)=d(i)−w ^(H)(i) r _(b)(i)  (20)

Based on the signal selection that minimizes |ζ_(b)(i)|², it is chosenthe corresponding reduced-rank observation vector r(i) and it isselected the error of the proposed iterative RLS algorithm ζ(i) as theerror ζ_(b)(i) with smallest squared magnitude of the B branches

$\begin{matrix}{{\zeta(i)} = {\arg\mspace{14mu}{\min\limits_{1 \leq b \leq B}{{\zeta_{b}(i)}}^{2}}}} & (21)\end{matrix}$

In order to compute parameter estimates, it is considered the timeaverage estimate of the matrix R(i), required in (11), given by

${{\hat{\overset{\_}{R}}(i)} = {\sum\limits_{j = 1}^{i}{\lambda^{i - j}{\overset{\_}{r}(j)}{{\overset{\_}{r}}^{H}(j)}}}},$where λ(0<λ≦1|) is the forgetting factor, that can be alternativelyexpressed by

${\hat{\overset{\_}{R}}(i)} = {{\lambda\;{\hat{\overset{\_}{R}}( {i - 1} )}} + {{\overset{\_}{r}(i)}{{{\overset{\_}{r}}^{H}(i)}.}}}$To avoid the inversion of

$\hat{\overset{\_}{R}}(i)$required in (11), the matrix inversion lemma is used and it is define

${P(i)} = {{\hat{\overset{\_}{R}}}^{- 1}(i)}$and the gain vector G(i) as

$\begin{matrix}{{G(i)} = \frac{\lambda^{- 1}{P( {i - 1} )}{\overset{\_}{r}(i)}}{1 + {\lambda^{- 1}{{\overset{\_}{r}}^{H}(i)}{P( {i - 1} )}{\overset{\_}{r}(i)}}}} & (22)\end{matrix}$and thus P(i) can be rewritten asP(i)=λ⁻¹ P(i−1)−λ⁻¹ G(i) r ^(H)(i)P(i−1)  (23)

By rearranging (23), G(i)=λ⁻¹P(i−1) r(i)−λ⁻¹G(i) r ^(H)(i)P(i−1)r(i)=P(i) r(i). By employing the LS solution (a time average of (11))and the recursion

${\hat{\overset{\_}{p}}(i)} = {{\lambda\;{\hat{\overset{\_}{p}}( {i - 1} )}} + {{d^{*}(i)}{\overset{\_}{r}(i)}}}$it is obtained

$\begin{matrix}{{w(i)} = {{{{\hat{\overset{\_}{R}}}^{- 1}(i)}{\hat{\overset{\_}{p}}(i)}} = {{\lambda\;{P(i)}{\hat{\overset{\_}{p}}( {i - 1} )}} + {{P(i)}{\overset{\_}{r}(i)}{d^{*}(i)}}}}} & (24)\end{matrix}$

Substituting (23) into (24) yields:w(i)=w(i−1)+G(i)ζ*(i)  (25)where the a priori estimation error is described by ζ(i)=d(i)−w^(H)(i)r(i). Similar recursions for the interpolator are devised by using (12).The estimate

${\hat{\overset{\_}{R}}}_{u}(i)$can be obtained through

${{\hat{\overset{\_}{R}}}_{u}(i)} = {\sum\limits_{j = 1}^{i}{\lambda^{i - j}{u(j)}{u^{H}(j)}}}$and can be alternatively written as

${{\hat{\overset{\_}{R}}}_{u}(i)} = {{\lambda\;{{\hat{\overset{\_}{R}}}_{u}( {i - 1} )}} + {{u(i)}{{u^{H}(i)}.}}}$To avoid the inversion of

${\hat{\overset{\_}{R}}}_{u}(i)$the matrix inversion lemma is used and again for convenience ofcomputation it is defined

${P_{u}(i)} = {{\hat{\overset{\_}{R}}}_{u}^{- 1}(i)}$and the Kalman gain vector G_(all (i) as:)

$\begin{matrix}{{G_{u}(i)} = \frac{\lambda^{- 1}{P_{u}( {i - 1} )}{u(i)}}{1 + {\lambda^{- 1}{u^{H}(i)}{P_{u}( {i - 1} )}{u(i)}}}} & (26)\end{matrix}$and thus rewriting (26) asP _(u)(i)=λ⁻¹ P _(u)(i−1)−λ⁻¹ G _(u)(i)u ^(H)(i)P _(u)(i−1)  (27)

By processing in a similar approach to the one taken to obtain (25) itis arrived atv(i)=v(i−1)+G _(u)(i)ζ*(i)  (28)

The RLS algorithm for the proposed structure trades off a computationalcomplexity of O(M²) against two RLS algorithms operating in parallel,with complexity O((M/L)²) and O(N_(I) ²), respectively. Because N_(I) issmall (N_(I)<<M and M/L<<M, as M/L and N_(I) do not scale with systemsize) the computational advantage of the RLS combined with the INTstructure is rather significant. It should be remarked that fast RLSversions of the proposed scheme are possible due to the absence of thetime-shifting properties required by those fast techniques. In TableIII, it is illustrated the pseudo-code of the algorithm described here.

TABLE III RLS iterative algorithm for the design of proposedreduced-rank filtering scheme. Algorithm III: Initialize w(0) = [0 . . .0]^(T) and v(0) = [1 0 . . . 0]^(T) Choose parameters B, L andforgetting factor λ for each time instant (i) do Compute M × 1 vectorr_(I) (i) = V^(H) (i)r(i) =

_(o) (i)v* (i) Obtain M/L × 1 vectors r _(b) (i) = D_(b)r_(I) (i) forthe B branches Select D_(b) that minimizes |ζ_(b)(i)|² (D_(b) becomesD(i)) Obtain r(i) and compute P(i) and G(i) Obtain u(i) and computeP_(u)(i) and G_(u)(i) Compute v(i) = v(i−1)+G_(u) (i)ζ* (i) and w(i) =w(i−1)+G(i)ζ* (i) Reduced-rank sample estimate: {circumflex over (d)}(i)= w^(H) (i)r(i)

1. A system for adaptive reduced-rank parameter estimation using anadaptive decimation and interpolation scheme comprising: (a) a firstfinite impulse response (FIR) interpolator filter with N_(I) weights atthe front-end of the system, represented by the N_(I)×1 parameter vectorv(i)=[v₀ ^((i)) . . . v_(NI) ^((i))]^(T) that filters the inputdiscrete-time signal r(i) with dimension M×1 and results in theinterpolated received vector r_(I)(i) with dimension M×1; (b) anadaptive decimation unit that has B parallel branches with appropriatedecimation patterns represented by the M/L×M matrix D_(b) that isresponsible for the dimensionally reduction of the M×1 interpolatedreceived vector r_(I)(i), resulting in B reduced-rank decimated vectorsr _(b)(i) with dimension M/L×1; (c) a signal selection scheme thatchooses the decimation pattern D_(b) that minimizes the squared norm ofthe error signal defined by e_(b)(i)=d(i)−w^(H)(i) r _(b)(i), where d(i)is a reference signal, the selection scheme utilizes the correspondingM/L×1 vector r _(b)(i) that yields the minimum |e_(b)(i)|² r(i) for theremaining signal processing; and (d) a second FIR filter with M/Lweights, denoted reduced-rank filter w(i)=[w₀ ^((i)) . . . w_(M/L)^((i))]^(T), that is responsible for the estimation of the desiredsignal, the system computes the inner product of w(i) with r(i) in orderto yield the parameter estimation for time instant (i) given byd_(est)(i)=w^(H)(i)r(i).
 2. The system of claim 1, wherein an optimaldecimation scheme that is based on the extensive search of all possibledecimation patterns and corresponds to B=(MI)/((M−M/L)I) decimationpatterns and branches.
 3. The system of claim 1, wherein a suboptimaldecimation scheme that is based on B independent and random selecteddecimation patterns.
 4. The system of claim 1, wherein a suboptimaldecimation scheme that is based on B pre-stored decimation patterns thatcorrespond to a uniform scheme whose index of samples chosen at each rowm is shifted by one position for each branch b.
 5. The system of claim1, wherein interpolator v(i) and reduced-rank w(i) filters are computedthrough a joint optimization of both parameter vectors.
 6. The system ofclaim 1, wherein the process of converting analog to discrete-timesignals is accomplished by standard analog-to-digital converters.
 7. Thesystem of claim 5, wherein the joint parameter optimization of theinterpolator v(i) and reduced-rank w(i) filters is carried out bystandard adaptive estimation approaches such as the LMS and the RLS. 8.A method for processing discrete-time signals organized into datavectors and reducing the number of elements for parameter estimationcomprising the steps of (a) pre-filtering the observed data vector withdimension M×1 with a first time-varying FIR filter with order N_(I)−1;(b) discarding M−M/L samples of the M×1 pre-filtered vector in Bdifferent ways, yielding B candidate vectors with reduced dimension M/L;(c) computing the inner product of the B candidate vectors r _(b)(i)with a second time-varying FIR filter w(i), with M/L weights, yielding Bcandidate scalar estimates d_(b)(i)=w^(H)(i) r _(b)(i); (d) calculatingthe B error signals e_(b)(i) between the desired signal d(i) and the Bcandidates d_(est b)(i); (e) selecting the error signal e_(b)(i) withsmallest magnitude and its corresponding candidate vector r_bar_(b)(i),the selected candidate vector r_bar_(b)(i) and error e_(b)(i) become thevector r(i) and the error e(i) to be processed, respectively; (f)computing the inner product of the selected vector r(i) with the secondtime-varying FIR filter with M/L weights, which correspond to thedesired signal estimate d_(est)(i)=w^(H)(i) r _(b)(i); and (g) adjustingthe weights of the first and second time-varying FIR filters with ajoint optimization procedure.
 9. The method of claim 8, wherein the stepof adaptively processing the received signals comprises of collecting,storing the samples in vectors and matrices and performing standardmathematical operations such as additions, multiplications, subtractionsand divisions.
 10. The method of claim 8, wherein the step ofpre-processing the received signals comprises the steps of: (a)analog-to-digital converting; (b) down converting the received signalsto baseband; and (c) applying a filter matched to the pulse used in theapplication to the converted received signals.
 11. The method of claim8, further comprising the step of creating a data vector from snapshotsof the received signals over time.